Reactor steady-state approximation

Coupled Reactions in Flow Reactors The Steady-state Approximation 41... 

Assuming that the catalytic reaction takes place in a flow reactor under stationary conditions, we may use the steady state approximation to eliminate the fraction of adsorbed intermediate from the rate expressions to yield ... 

In industry, as well as in a test reactor in the laboratory, we are most often interested in the situation where a constant flow of reactants enters the reactor, leading to a constant output of products. In this case all transient behavior due to start up phenomena have died out and coverages and rates have reached a constant value. Hence, we can apply the steady state approximation, and set all differentials in Eqs. (142)-(145) equal to zero ... 

Equation (48) e ees with experimental results in some circumstances. This does not mean the mechanism is necessarily correct. Other mechanisms may be compatible with the experimental data and this mechanism may not be compatible with experiment if the physical conditions (temperature and pressure etc.) are changed. Edelson and Allara [15] discuss this point with reference to the application of the steady state approximation to propane pyrolysis. It must be remembered that a laboratory study is often confined to a narrow range of conditions, whereas an industrial reactor often has to accommodate large changes in concentrations, temperature and pressure. Thus, a successful kinetic model must allow for these conditions even if the chemistry it portrays is not strictly correct. One major problem with any kinetic model, whatever its degree of reality, is the evaluation of the rate cofficients (or model parameters). This requires careful numerical analysis of experimental data it is particularly important to identify those parameters to which the model predictions are most sensitive. 

Textbooks state that the pseudo-steady-state approximation will be valid if the concentration of a species is small. However, one then proceeds by setting its time derivative equal to zero (]/t/f = 0) in the batch reactor equation, not by setting the concentration (CH3CO ) equal to zero. This logic is not obvious from the batch reactor equations because setting the derivative of a concentration equal to zero is not the same as setting its concentration equal to zero. 

The steady state approximation eliminates transient behavior in the kinetics. However, it is only the transient behavior of the rates and coverages that has been eliminated. The expression for the rate obtained through the steady state approximation is perfectly suitable for the simulation of e g the conversion through a catalyst bed or most aspects of the transient behavior of a reactor. 

Applying the rate expressions to Equations 1-222, 1-223, 1-224, 1-225 and 1-226, and using the steady state approximation for CH3, C2H5, and H, for a constant volume batch reactor yields ... 

Some work has already been done on the simulation of transient behavior of moving bed coal gasifiers. However, the analysis is not based on the use of a truly dynamic model but instead uses a steady state gasifier model plus a pseudo steady state approximation. For this type of approach, the time response of the gasifier to reactor input changes appears as a continuous sequence of new steady states. 

One further note, the University of Delaware gasifier model used in the pseudo steady state approximation assumes that the gas and solids temperatures are the same within the reactor. That assumption removes an important dynamic feedback effect between the countercurrent flowing gas and solids streams. This is particularly important when the burning zone moves up and down within the reactor in an oscillatory manner in response to a step change in operating conditions. 

The condition expressed by the Bodenstein approximation rx = 0 is often misleadingly called a steady state. It is not. It is not a time-independent state, only a state in which a specific variation with time (or reactor space time) is small compared with the others. In fact, some older textbooks applied what they called the steady-state approximation to batch reactions in order to derive the time dependence of the concentrations, unwittingly leading the incorrect presumption of a steady state ad absurdum. And a continuous stirred-tank or tubular reactor may, and usually does, come to a true steady state, even if the Bodenstein approximation is and remains inapplicable. [The approximation compares process rates r, it is irrelevant for its validity whether or not the reactor comes to a steady state, that is, whether the rates of change, dC /dr, become zero.]... 

Expressions for relative concentrations of the intermediates and products are developed from concepts discussed in earlier chapters, namely, the use of a total catalytic site balance and the application of the steady-state approximation. The total amount of rhodium catalyst in the reactor is considered constant, [ ]q, so that the site balance becomes ... 

Quasi-steady-state periodic regime (T Tj. The input variable varies rather slowly compared to the dynamics of the system, and the system follows the input variable almost exactly. The time-averaged performance of the reactor is calculated applying the quasi-steady-state approximation to the state of the system and averaging out the resulting performances at any time. 

Assuming that the steady-state approximation holds for [M A ], and that reactant and helium are both in large excess, integrating over the residence time t in the reactor yields an expression for the reactivity R ... 

Note that the steady-state approximation does not imply that [X] is even approximately constant, only that its absolute rate of change is very much smaller than that of [A] and [D], Since according to the reaction scheme d[D]/dt = 2[X][C], the assumption that [X] is constant would lead—for the case in which C is in large excess—to the absurd conclusion that formation of the product D will continue at a constant rate even after the reactant A has been consumed. (2) In a stirred-flow reactor, a steady state implies a regime so that all concentrations are independent of time. 

The optimal catalyst distribution problem was studied in an adiabatic reactor (Ogunye and Ray, 197la,b). The optimal initial distribution of catalyst activity along the axis of a tubular fixed-bed reactor was examined for a class of reactivation-deactivation problems by Gryaert and Crowe (1976). A general set of simultaneous reactions was considered, quasi steady state approximation was used, and the decay of the catalyst expressed as a function of temperature, concentration and catalyst activity. The influence of various initial catalyst activity distributions upon the reactor performance was also considered. 

The reactor time scale is much shorter than the time scale for catalyst decay so that the quasi-steady state approximation is valid. 

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